The existence of solution of a class of two-order quasilinear boundary value problem

Abstract

Ref. [1] discussed the existence of positive solutions of quasilinear two-point boundary problems:

$$\left\{ \begin{gathered} \frac{{d^2 y}}{{dt^2 }} + y^\alpha + y^\beta = 0, 0 \leqslant t \leqslant 1 \hfill \\ y(0) = y'(1) = 0 \hfill \\ \end{gathered} \right.$$

but it restricts 0<α<1<β. This article proves the existence of positive solution to this equation:

$$\left\{ \begin{gathered} \frac{{d^2 x}}{{dt^2 }} + x^\alpha + 1 = 0, 0 \leqslant t \leqslant 1, \alpha > 0 \hfill \\ x(0) = x'(1) = 0 \hfill \\ \end{gathered} \right.$$